GENERALIZED CLASSES OF ANALYTIC FUNCTIONS DEFINED BY Q-CALCULUS

Abstract

This thesis introduces a new differential operator that involves q-calculus. The operator is derived using Hadamard product between the q-analogue of the Ruscheweyh derivative by Aldweby and Darus with the symmetric Salagean differential operator by Ibrahim and Darus. The combination of the q-analogue of the Ruscheweyh derivative and the symmetric Salagean differential operator in this thesis motivates the development of a new differential operator, offering a new perspective on mathematical analysis that has not been investigated previously. This operator is employed to introduce new classes of univalent functions, particularly those with a positive first derivative, such as starlike, convex, uniformly convex, uniformly starlike, and α–close-to-convex functions. The thesis analyzes the geometrical properties of these functions, including the coefficient problem, distortion theorem, and extreme value. One of the primary focuses of this thesis is the Fekete-Szegö problem, which addresses the location of zeros in polynomials with certain restrictions on their coefficients. The research generalizes previous findings and emphasizes the significance of the Hankel matrix in complex analysis, providing insights into the relationship between the Fekete-Szegö problem and the geometry of univalent functions. For this issue, a generalized quantum calculus operator and Hadamard products are utilized to define various classes of univalent functions. Furthermore, two classes of uniformly geometric functions connected with the differential operator aforementioned are introduced and their properties are investigated such as coefficient problem, and distortion theorem. In addition, a new generalized complex q-fractional integral operator is introduced, derived from a successive application of the operator of fractional integration defined by Srivastava and Owa in 1989 and the q-Ruscheweyh derivative operator introduced by Aldweby and Darus in 2014. The thesis examines a coefficient problem involving the second Hankel determinant H2(2) for some normalized families of q-starlike and q-convex functions in the open unit disk. These families are defined using the new q-fractional integral operator, offering a new approach to the study of analytic and univalent functions. In summary, this thesis contributes significantly to complex geometric theory by introducing new generalized quantum calculus and q-fractional integral operators and exploring their properties. It also introduces several new classes of univalent functions and analyzes their geometrical properties, including the Fekete-Szegö problem, the class of uniformly starlike functions, and the class of uniformly convex functions. The new generalized complex q-fractional integral operator is investigated in the context of q-convex and q-starlike functions, offering new directions for further exploration in complex geometric theory.